What are the divisors of 1617?

1, 3, 7, 11, 21, 33, 49, 77, 147, 231, 539, 1617

There is a total of 12 positive divisors.
The sum of these divisors is 2736.
The arithmetic mean is 228.

12 odd divisors

1, 3, 7, 11, 21, 33, 49, 77, 147, 231, 539, 1617

How to compute the divisors of 1617?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 1617 by each of the numbers from 1 to 1617 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

1617 / 1 = 1617 (the remainder is 0, so 1 is a divisor of 1617)
1617 / 2 = 808.5 (the remainder is 1, so 2 is not a divisor of 1617)
1617 / 3 = 539 (the remainder is 0, so 3 is a divisor of 1617)
...
1617 / 1616 = 1.0006188118812 (the remainder is 1, so 1616 is not a divisor of 1617)
1617 / 1617 = 1 (the remainder is 0, so 1617 is a divisor of 1617)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 1617 (i.e. 40.211938525766). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

1617 / 1 = 1617 (the remainder is 0, so 1 and 1617 are divisors of 1617)
1617 / 2 = 808.5 (the remainder is 1, so 2 is not a divisor of 1617)
1617 / 3 = 539 (the remainder is 0, so 3 and 539 are divisors of 1617)
...
1617 / 39 = 41.461538461538 (the remainder is 18, so 39 is not a divisor of 1617)
1617 / 40 = 40.425 (the remainder is 17, so 40 is not a divisor of 1617)